The electrical engineering handbook
Tallarida, R.J. “Section XII – Mathematics, Symbols, and Physical Constants” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 The mathematical equation used to generate this three-dimensional figure is worth a thousand words. It represents a single-solitron surface for the sine-Gordon equation w uv = sin w. Among the areas in which the sine-Gordon equation arises is that of wave propagation on nonlinear transmission lines and in semi-conductors. The equation is famous because it is known to have a nonlinear superposition principle obtainable by means of a Bäcklund Transformation. The sine-Gordon equation is an example of an evolution equation which has an infinite sequence of non-trivial conservation laws so important in the fields of engineering and physics. For further information on the Bäcklund Transformation see Bäck- lund Transformations and their Application, Rogers and Shadwick, Academic Press, 1982. This three-dimensional projection was generated using the MAPLE® software package. MAPLE® is one of three important mathematical computer packages that offer a variety of analytical and numerical software for use by scientists, engineers, and mathematicians. This figure was developed by W.K. Schief and C. Rogers and the Center for Dynamical Systems and Nonlinear Studies at Georgia Institute of Technology and the University of New South Wales in Sydney, Australia. (Figure courtesy of Schief and Rogers.) © 2000 by CRC Press LLC © 2000 by CRC Press LLC XII Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) Definitions of SI Base Units•Names and Symbols for the SI Base Units•SI Derived Units with Special Names and Symbols•Units in Use Together with the SI Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples•Conversion Factors—Metric to English• Conversion Factors—English to Metric•Conversion Factors—General•Temperature Factors• Conversion of Temperatures Physical Constants General• p Constants•Constants Involving e• Numerical Constants Symbols and Terminology for Physical and Chemical Quantities Classical Mechanics•Electricity and Magnetism•Electromagnetic Radiation•Solid State Credits Ronald J. Tallarida Temple University HE GREAT ACHIEVEMENTS in engineering deeply affect the lives of all of us and also serve to remind us of the importance of mathematics. Interest in mathematics has grown steadily with these engineering achievements and with concomitant advances in pure physical science. Whereas scholars in nonscien- tific fields, and even in such fields as botany, medicine, geology, etc., can communicate most of the problems and results in nonmathematical language, this is virtually impossible in present-day engineering and physics. Yet it is interesting to note that until the beginning of the twentieth century engineers regarded calculus as something of a mystery. Modern students of engineering now study calculus, as well as differential equations, complex variables, vector analysis, orthogonal functions, and a variety of other topics in applied analysis. The study of systems has ushered in matrix algebra and, indeed, most engineering students now take linear algebra as a core topic early in their mathematical education. This section contains concise summaries of relevant topics in applied engineering mathematics and certain key formulas, that is, those formulas that are most often needed in the formulation and solution of engineering problems. Whereas even inexpensive electronic calculators contain tabular material (e.g., tables of trigonometric and logarithmic functions) that used to be needed in this kind of handbook, most calculators do not give symbolic results. Hence, we have included formulas along with brief summaries that guide their use. In many cases we have added numerical examples, as in the discussions of matrices, their inverses, and their use in the solutions of linear systems. A table of derivatives is included, as well as key applications of the derivative in the solution of problems in maxima and minima, related rates, analysis of curvature, and finding approximate T © 2000 by CRC Press LLC roots by numerical methods. A list of infinite series, along with the interval of convergence of each, is also included. Of the two branches of calculus, integral calculus is richer in its applications, as well as in its theoretical content. Though the theory is not emphasized here, important applications such as finding areas, lengths, volumes, centroids, and the work done by a nonconstant force are included. Both cylindrical and spherical polar coordinates are discussed, and a table of integrals is included. Vector analysis is summarized in a separate section and includes a summary of the algebraic formulas involving dot and cross multiplication, frequently needed in the study of fields, as well as the important theorems of Stokes and Gauss. The part on special functions includes the gamma function, hyperbolic functions, Fourier series, orthogonal functions, and both Laplace and z -transforms. The Laplace transform provides a basis for the solution of differential equations and is fundamental to all concepts and definitions underlying analytical tools for describing feedback control systems. The z -transform, not discussed in most applied mathematics books, is most useful in the analysis of discrete signals as, for example, when a computer receives data sampled at some prespecified time interval. The Bessel functions, also called cylindrical functions, arise in many physical applications, such as the heat transfer in a “long” cylinder, whereas the other orthogonal functions discussed—Legendre, Hermite, and Laguerre polynomials—are needed in quantum mechanics and many other subjects (e.g., solid-state electronics) that use concepts of modern physics. The world of mathematics, even applied mathematics, is vast. Even the best mathematicians cannot keep up with more than a small piece of this world. The topics included in this section, however, have withstood the test of time and, thus, are truly core for the modern engineer. This section also incorporates tables of physical constants and symbols widely used by engineers. While not exhaustive, the constants, conversion factors, and symbols provided will enable the reader to accommodate a majority of the needs that arise in design, test, and manufacturing functions. © 2000 by CRC Press LLC Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) The International System of units (SI) was adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960. It is a coherent system of units built form seven SI base units, one for each of the seven dimensionally independent base quantities: they are the meter, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given below. The SI derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity. In the International System there is only one SI unit for each physical quantity. This is either the appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units. It is recommended that only SI units be used in science and technology (with SI prefixes where appropriate). Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units. This section is based on information supplied by IUPAC. Definitions of SI Base Units Meter —The meter is the length of path traveled by light in vacuum during a time interval of 1/299 792 458 of a second (17th CGPM, 1983). Kilogram —The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (3rd CGPM, 1901). Second —The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM, 1967). Ampere —The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 ´ 10 –7 newton per meter of length (9th CGPM, 1948). Greek Greek English Greek Greek English letter name equivalent letter name equivalent Aa Alpha a Nn Nu n Bb Beta b Xx Xi x Gg Gamma g Oo Omicron ˘ o Dd Delta d Pp Pi p Ee Epsilon ˘ e Rr Rho r Zz Zeta z Ss Sigma s Hh Eta e – Tt Tau t QqJ Theta th Uu Upsilon u Ii Iota i Ffj Phi ph Kk Kappa k Cc Chi ch Ll Lambda l Yy Psi ps Mm Mu m Ww Omega o – © 2000 by CRC Press LLC Kelvin —The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (13th CGPM, 1967). Mole —The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, or other particles, or specified groups of such particles (14th CGPM, 1971). Examples of the use of the mole: 1 mol of H 2 contains about 6.022 ´ 10 23 H 2 molecules, or 12.044 ´ 10 23 H atoms 1 mol of HgCl has a mass of 236.04 g 1 mol of Hg 2 Cl 2 has a mass of 472.08 g 1 mol of Hg 2 2+ has a mass of 401.18 g and a charge of 192.97 kC 1 mol of Fe 0.91 S has a mass of 82.88 g 1 mol of e – has a mass of 548.60 m g and a charge of –96.49 kC 1 mol of photons whose frequency is 10 14 Hz has energy of about 39.90 kJ Candela —The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 ´ 10 12 hertz and that has a radiant intensity in that direction of (1/683) watt per steradian (16th CGPM, 1979). Names and Symbols for the SI Base Units SI Derived Units with Special Names and Symbols Physical quantity Name of SI unit Symbol for SI unit length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd Name of Symbol for Expression in terms of SI base unitsPhysical quantity SI unit SI unit frequency 1 hertz Hz s –1 force newton N m kg s –2 pressure, stress pascal Pa N m –2 =m –1 kg s –2 energy, work, heat joule J N m = m 2 kg s –2 power, radiant flux watt W J s –1 = m 2 kg s –3 electric charge coulomb C A s electric potential, volt V J C –1 = m 2 kg s –3 A –1 electromotive force electric resistance ohm W V A –1 = m 2 kg s –3 A –2 electric conductance siemens S W –1 = m –2 kg –1 s 3 A 2 electric capacitance farad F C V –1 = m –2 kg –1 s 4 A 2 magnetic flux density tesla T V s m –2 = kg s –2 A –1 magnetic flux weber Wb V s = m 2 kg s –2 A –1 inductance henry H V A –1 s= m 2 kg s –2 A –2 Celsius temperature 2 degree Celsius ° CK luminous flux lumen lm cd sr illuminance lux lx cd sr m –2 activity (radioactive) becquerel Bq s –1 absorbed dose (of radiation) gray Gy J kg –1 = m 2 s –2 dose equivalent sievert Sv J kg –1 = m 2 s –2 (dose equivalent index) © 2000 by CRC Press LLC Units in Use Together with the SI These units are not part of the SI, but it is recognized that they will continue to be used in appropriate contexts. SI prefixes may be attached to some of these units, such as milliliter, ml; millibar, mbar; megaelectronvolt, MeV; kilotonne, ktonne. Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples plane angle radian rad 1 = m m –1 solid angle steradian sr 1 = m 2 m –2 1 For radial (circular) frequency and for angular velocity the unit rad s –1 , or simply s –1 , should be used, and this may not be simplified to Hz. The unit Hz should be used only for frequency in the sense of cycles per second. 2 The Celsius temperature q is defined by the equation: q / ° C = T /K – 273.15 The SI unit of Celsius temperature interval is the degree Celsius, ° C, which is equal to the kelvin, K. ° C should be treated as a single symbol, with no space between the ° sign and the letter C. (The symbol °K, and the symbol °, should no longer be used.) Physical Symbol Value in SI unitsquantity Name of unit for unit time minute min 60 s time hour h 3600 s time day d 86 400 s plane angle degree ° (p/180) rad plane angle minute ¢ (p/10 800) rad plane angle second ² (p/648 000) rad length ångstrom 1 Å10 –10 m area barn b 10 –28 m 2 volume litre l, L dm 3 = 10 –3 m 3 mass tonne t Mg = 10 3 kg pressure bar 1 bar 10 5 Pa = 10 5 N m –2 energy electronvolt 2 eV (= e ´ V) »1.60218 ´ 10 –19 J mass unified atomic mass unit 2,3 u (=m a ( 12 C)/12) »1.66054 ´ 10 –27 kg 1 The ångstrom and the bar are approved by CIPM for “temporary use with SI units,” until CIPM makes a further recommendation. However, they should not be introduced where they are not used at present. 2 The values of these units in terms of the corresponding SI units are not exact, since they depend on the values of the physical constants e (for the electronvolt) and N a (for the unified atomic mass unit), which are determined by experiment. 3 The unified atomic mass unit is also sometimes called the dalton, with symbol Da, although the name and symbol have not been approved by CGPM. Multiples and Multiples and submultiples Prefixes Symbols submultiples Prefixes Symbols 10 18 exa E 10 –1 deci d 10 15 peta P 10 –2 centi c 10 12 tera T 10 –3 milli m 10 9 giga G 10 –6 micro m (Greek mu) Name of Symbol for Expression in terms of SI base unitsPhysical quantity SI unit SI unit © 2000 by CRC Press LLC Conversion Factors—Metric to English Conversion Factors—English to Metric* Conversion Factors—General* 10 6 mega M 10 –9 nano n 10 3 kilo k 10 –12 pico p 10 2 hecto h 10 –15 femto f 10 deca da 10 –18 atto a To obtain Multiply By Inches Centimeters 0.3937007874 Feet Meters 3.280839895 Yards Meters 1.093613298 Miles Kilometers 0.6213711922 Ounces Grams 3.527396195 ´ 10 –2 Pounds Kilogram 2.204622622 Gallons (U.S. Liquid) Liters 0.2641720524 Fluid ounces Milliliters (cc) 3.381402270 ´ 10 –2 Square inches Square centimeters 0.155003100 Square feet Square meters 10.76391042 Square yards Square meters 1.195990046 Cubic inches Milliliters (cc) 6.102374409 ´ 10 –2 Cubic feet Cubic meters 35.31466672 Cubic yards Cubic meters 1.307950619 To obtain Multiply By Microns Mils 25.4 Centimeters Inches 2.54 Meters Feet 0.3048 Meters Yards 0.9144 Kilometers Miles 1.609344 Grams Ounces 28.34952313 Kilograms Pounds 0.45359237 Liters Gallons (U.S. Liquid) 3.785411784 Millimeters (cc) Fluid ounces 29.57352956 Square centimeters Square inches 6.4516 Square meters Square feet 0.09290304 Square meters Square yards 0.83612736 Milliliters (cc) Cubic inches 16.387064 Cubic meters Cubic feet 2.831684659 ´ 10 –2 Cubic meters Cubic yards 0.764554858 To obtain Multiply By Atmospheres Feet of water @ 4°C 2.950 ´ 10 –2 Atmospheres Inches of mercury @ 0°C 3.342 ´ 10 –2 Atmospheres Pounds per square inch 6.804 ´ 10 –2 BTU Foot-pounds 1.285 ´ 10 –3 BTU Joules 9.480 ´ 10 –4 Cubic feet Cords 128 Degree (angle) Radians 57.2958 Ergs Foot-pounds 1.356 ´ 10 7 Multiples and Multiples and submultiples Prefixes Symbols submultiples Prefixes Symbols *Boldface numbers are exact; others are given to ten significant figures where so indicated by the multiplier factor. © 2000 by CRC Press LLC Temperature Factors °F = 9/5 (°C) + 32 Fahrenheit temperature = 1.8 (temperature in kelvins) – 459.67 °C = 5/9 [(°F) – 32)] Celsius temperature = temperature in kelvins – 273.15 Fahrenheit temperature = 1.8 (Celsius temperature) + 32 Conversion of Temperatures Physical Constants General Equatorial radius of the earth = 6378.388 km = 3963.34 miles (statute). Polar radius of the earth, 6356.912 km = 3949.99 miles (statute). 1 degree of latitude at 40° = 69 miles. 1 international nautical mile = 1.15078 miles (statute) = 1852 m = 6076.115 ft. Mean density of the earth = 5.522 g/cm 3 = 344.7 lb/ft 3 Constant of gravitation (6.673 ± 0.003) ´ 10 –8 cm 3 gm –1 s –2 . Feet Miles 5280 Feet of water @ 4°C Atmospheres 33.90 Foot-pounds Horsepower-hours 1.98 ´ 10 6 Foot-pounds Kilowatt-hours 2.655 ´ 10 6 Foot-pounds per min Horsepower 3.3 ´ 10 4 Horsepower Foot-pounds per sec 1.818 ´ 10 –3 Inches of mercury @ 0°C Pounds per square inch 2.036 Joules BTU 1054.8 Joules Foot-pounds 1.35582 Kilowatts BTU per min 1.758 ´ 10 –2 Kilowatts Foot-pounds per min 2.26 ´ 10 –5 Kilowatts Horsepower 0.745712 Knots Miles per hour 0.86897624 Miles Feet 1.894 ´ 10 –4 Nautical miles Miles 0.86897624 Radians Degrees 1.745 ´ 10 –2 Square feet Acres 43560 Watts BTU per min 17.5796 From To °Celsius °Fahrenheit t F = (t C ´ 1.8) + 32 Kelvin T K = t C + 273.15 °Rankine T R = (t C + 273.15) ´ 18 °Fahrenheit °Celsius t C = Kelvin T k = + 273.15 °Rankine T R = t F + 459.67 Kelvin °Celsius t C = T K – 273.15 °Rankine T R = T K ´ 1.8 °Rankine Kelvin T K = °Farenheit t F = T R – 459.67 To obtain Multiply By t F 32– 1.8 ---------------- t F 32– 1.8 ---------------- T R 1.8 ------- © 2000 by CRC Press LLC Acceleration due to gravity at sea level, latitude 45° = 980.6194 cm/s 2 = 32.1726 ft/s 2 . Length of seconds pendulum at sea level, latitude 45° = 99.3575 cm = 39.1171 in. 1 knot (international) = 101.269 ft/min = 1.6878 ft/s = 1.1508 miles (statute)/h. 1 micron = 10 –4 cm. 1 ångstrom = 10 –8 cm. Mass of hydrogen atom = (1.67339 ± 0.0031) ´ 10 –24 g. Density of mercury at 0°C = 13.5955 g/ml. Density of water at 3.98°C = 1.000000 g/ml. Density, maximum, of water, at 3.98°C = 0.999973 g/cm 3 . Density of dry air at 0°C, 760 mm = 1.2929 g/l. Velocity of sound in dry air at 0°C = 331.36 m/s – 1087.1 ft/s. Velocity of light in vacuum = (2.997925 ± 0.000002) ´ 10 10 cm/s. Heat of fusion of water 0°C = 79.71 cal/g. Heat of vaporization of water 100°C = 539.55 cal/g. Electrochemical equivalent of silver 0.001118 g/s international amp. Absolute wavelength of red cadmium light in air at 15°C, 760 mm pressure = 6438.4696 Å. Wavelength of orange-red line of krypton 86 = 6057.802 Å. p Constants p = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511 1/p = 0.31830 98861 83790 67153 77675 26745 02872 40689 19291 48091 p 2 = 9.8690 44010 89358 61883 44909 99876 15113 53136 99407 24079 log e p = 1.14472 98858 49400 17414 34273 51353 05871 16472 94812 91531 log 10 p = 0.49714 98726 94133 85435 12682 88290 89887 36516 78324 38044 log 10 ߛ2p = 0.39908 99341 79057 52478 25035 91507 69595 02099 34102 92128 Constants Involving e e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996 1/e = 0.36787 94411 71442 32159 55237 70161 46086 74458 11131 03177 e 2 = 7.38905 60989 30650 22723 04274 60575 00781 31803 15570 55185 M = log 10 e = 0.43429 44819 03251 82765 11289 18916 60508 22943 97005 80367 1/M·=log e 10 = 2.30258 50929 94045 68401 79914 54684 36420 67011 01488 62877 log 10 M = 9.63778 43113 00536 78912 29674 98645 –10 Numerical Constants ߛ2 = 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37695 3ߛ2 = 1.25992 10498 94873 16476 72106 07278 22835 05702 51464 70151 log e 2 = 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 36026 log 10 2 = 0.30102 99956 63981 19521 37388 94724 49302 67881 89881 46211 ߛ3 = 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81039 3 ߛ3 = 1.44224 95703 07408 38232 16383 10780 10958 83918 69253 49935 log e 3 = 1.09861 22886 68109 69139 52452 36922 52570 46474 90557 82275 log 10 3 = 0.47712 12547 19662 43729 50279 03255 11530 92001 28864 19070 Symbols and Terminology for Physical and Chemical Quantities Name Symbol Definition SI unit Classical Mechanics mass m kg reduced mass mm = m 1 m 2 /(m 1 + m 2 )kg density, mass density rr = M/V kg m –3 relative density dd = r/r q l surface density r A , r S r A = m/A kg m –2